How can we describe nature mathematically? What is the essence of the finite element method? Can the problems of fluid mechanics be solved numerically? MIT Professor of Mathematics Gilbert Strang sets up the debate.

But if we have a hundred thousand functions, they can be just maybe little hat functions, just up and down again, simple functions. Their combinations, if we have many, can give us close to the correct answer. The idea of the finite element method is a combination of Galerkin’s idea of test functions with the idea of simple functions, where the physics is simple, the equations stay simple, but you have many, many, many functions. And that’s what the computer is happy with.

Now a finite element code has thousands and thousands of lines. Companies specialize in preparing a code to solve physics problems, engineering problems. And you pay for their work in preparing the software, preparing the code. Mathematicians, their part has been and still is to understand what is going on, how to solve those many, many equations, how close is the solution to the original problem. It’s a team work.

All numerical methods are waiting to be improved. The finite elements came first for structural problems, and that’s a class of equations where nothing is going to move too far, where the bridge just moves a little bit, and it’s important to know how much. But think of the difference between that and fluid problems… So you could say, for solid mechanics we’re good, for fluid mechanics we have much work to do, for gas mechanics we have much, much work to do.